Polynomial and Exponential Spatial Decay Estimates in Elasticity

In this note we investigate the spatial behavior of the solutions of a combination of a hyperbolic system with an elliptic system. We consider a semi-infinite cylinder which is the union of two sub-cylinders. In one of them, we assume an elastodynamical problem and in the other an elastostatic problem. Both are coupled through an interface. It is known that the elastostatic problem and the elastodynamic problem have a fast decay (at least exponential). However, as their spatial behaviors are of different kind, it is not clear how this combination could be controlled in a similar way. We prove that the decay of solutions can be controlled in a polynomial way. We also describe how to obtain an upper bound for the amplitude term. We conclude the paper sketching the exponential decay behavior for the harmonic vibrations.